a diagram shows a cross section of a triangular prism.
AB=BC=7cm. BM=5cm
m is the midpoint of the line segment AC.
a) explain why angle AMB must be a right angle.
b) calculate angle BCM.
c) calculate area of the triangle
ABC is an isosceles triangle, so the altitude BM intersects AC in the middle.
cos BCM = 5/7
area = 1/2 AC * BM
a) To explain why angle AMB must be a right angle, let's consider the properties of a prism. A prism consists of two congruent parallel bases connected by lateral faces. In a triangular prism, one of the bases is a triangle, and the other base is a congruent triangle parallel to the first.
In this case, let's say that base ABC is the bottom triangle of the prism, and M is the midpoint of the line segment AC. Since M is the midpoint, it divides the triangle into two congruent triangles: AMB and CMB.
Now, let's focus on triangle AMB. Since AB = BC = 7 cm, we can conclude that triangle ABC is an isosceles triangle. In an isosceles triangle, the base angles (opposite the congruent sides) are also congruent.
Therefore, angle AMB is congruent to angle CMB, and since the two angles form a linear pair (they share a common side, BM), their sum must be 180 degrees. Thus, angle AMB must be a right angle, measuring 90 degrees.
b) To calculate angle BCM, we can use the fact that triangle ABC is isosceles. We know that AB = BC = 7 cm, and BM = 5 cm. Let's label the point where BM intersects the base AC as point X.
By drawing a perpendicular line from point X to BC, we create two right triangles: BCM and BXM. Since BM is the height of triangle ABC, we can use the property that the perpendicular drawn from the vertex angle of an isosceles triangle bisects the base.
Since AB = BC = 7 cm, BX must be half of BC, so BX = BC/2 = 7/2 = 3.5 cm. Now, we have the lengths of the two sides of triangle BCM - BC = 7 cm and XM = 3.5 cm.
Using the Pythagorean Theorem, we can find the length of CM (hypotenuse):
CM^2 = BC^2 - XM^2
CM^2 = 7^2 - 3.5^2
CM^2 = 49 - 12.25
CM^2 = 36.75
CM = √36.75
CM ≈ 6.06 cm
Now that we know the lengths of all three sides of triangle BCM, we can use the Law of Cosines to calculate angle BCM:
cos(BCM) = (BC^2 + CM^2 - BM^2) / (2 * BC * CM)
cos(BCM) = (7^2 + 6.06^2 - 5^2) / (2 * 7 * 6.06)
cos(BCM) = (49 + 36.7236 - 25) / (84.84)
cos(BCM) = 60.7236 / 84.84
cos(BCM) ≈ 0.716
BCM ≈ cos^(-1)(0.716)
BCM ≈ 43.5 degrees
Therefore, angle BCM measures approximately 43.5 degrees.
c) To calculate the area of the triangle, we can use the formula for the area of a triangle: Area = (1/2) * base * height.
In this case, we can consider triangle ABC as the base of the prism, since it is one of the congruent parallel bases. The height of the triangle is the line segment BM, which measures 5 cm.
Area = (1/2) * base * height
Area = (1/2) * AB * BM
Area = (1/2) * 7 * 5
Area = 35/2
Area = 17.5 square cm
Therefore, the area of the triangle is 17.5 square cm.